- Uber is a ridesharing service that services millions of customers each day
- Data were collected on individual rides in New York City from September 2014 to August 2015
- Drivers want to know how to make the most money
December 3, 2020
We are interested in answering the following questions:
\[Y(t) = f(t) + \epsilon(t) \ \text{where} \ \epsilon(t) \sim \mathcal{N}(0, \sigma^2)\]
\[\boldsymbol{Y} | \boldsymbol{f} \sim \mathcal{N}(\boldsymbol{f}, \sigma^2 I)\]
\[\boldsymbol{f} \sim \mathcal{N}(\boldsymbol{0}, K_f) \ \text{where} \ K_f(t, t') = \tau^2 k(t, t')\]
\[\begin{bmatrix}
\boldsymbol{Y} \\
\boldsymbol{f}
\end{bmatrix}
\sim
\mathcal{N}
\Bigg(
\begin{bmatrix}
\boldsymbol{0} \\
\boldsymbol{0}
\end{bmatrix}
,
\begin{bmatrix}
\sigma^2 I + K_f & K_f \\
K_f & K_f
\end{bmatrix}\Bigg)\]
\[\boldsymbol{f} | \boldsymbol{Y} \sim \mathcal{N}(K_f (\sigma^2 I + K_f)^{-1} \boldsymbol{Y}, K_f - K_f(\sigma^2 I + K_f)^{-1}K_f)\]
\[Y(t) = f_{long}(t) + f_{medium}(t) + f_{short}(t) + \epsilon(t)\]
\[f_{*}(t) \sim \mathcal{N}(\boldsymbol{0}, K_*(t,t'))\]
\[K_*(t,t') = \tau^2_* \text{exp}\Bigg(-\frac{1}{2 l^2_*} |t-t'|^2\Bigg)\]
\[Y(s,t) = f(s,t) + \epsilon(s, t)\]
\[\boldsymbol{Y} | \boldsymbol{f} \sim \mathcal{N}(\boldsymbol{f}, \sigma^2 I)\]
\[\boldsymbol{f} \sim \mathcal{N}(\boldsymbol{0}, K_S \otimes K_T)\]
\[K_S(s,s') = \tau^2_S \text{exp}\Bigg(-\frac{1}{2 l^2_S} ||s-s'||^2_2\Bigg)\]
\[K_T(t,t') = \tau^2_T \text{exp}\Bigg(-\frac{1}{2 l^2_S} |t-t'|^2\Bigg)\]